Prepotential and the Seiberg-Witten Theory

Abstract

Some basic facts about the prepotential in the SW/Whitham theory are presented. Consideration begins from the abstract theory of quasiclassical $\tau$-functions , which uses as input a family of complex spectral curves with a meromorphic differential $dS$, subject to the constraint $\partial dS/\partial(moduli)= \ holomorphic$, and gives as an output a homogeneous prepotential on extended moduli space. Then reversed construction is discussed, which is straightforwardly generalizable from spectral {\it curves} to certain complex manifolds of dimension $d >1$ (like $K3$ and $CY$ families). Finally, examples of particular $N=2$ SUSY gauge models are considered from the point of view of this formalism. At the end we discuss similarity between the $WP^{12}_{1,1,2,2,6}$ -\-Calabi-\-Yau model with $h_{21}=2$ and the $1d$ $SL(2)$ Calogero/Ruijsenaars model, but stop short of the claim that they belong to the same Whitham universality class beyond the conifold limit.